Two games are strategically equivalent when you can relabel each player’s strategies with a one-to-one, onto mapping and, for every resulting strategy profile, the required outcomes and payoff or preference relationships are preserved. Matching the number of strategies, a few matrix entries, or even an equilibrium is not enough. If move order, information sets, or observations matter, compare the extensive-form structures as well, because a normal form can omit those features.
What “strategically equivalent” means
“Equivalent” is not a single test used for every game-theory question. Start by naming the game representation and the feature you need to preserve. In a finite strategic (normal-form) game, the basic objects are the players, each player’s strategy set, the possible outcomes, an outcome function, and the players’ preferences over outcomes. Numerical utility functions can represent those preferences, but utility numbers and preferences are not automatically the same equivalence criterion.
| # | Preview | Product | Price | |
|---|---|---|---|---|
| 1 |
|
Game Theory 101: The Complete Textbook | $14.99 | Buy on Amazon |
| 2 |
|
Game Theory: An Introduction | $48.43 | Buy on Amazon |
| 3 |
|
Game Theory | $65.57 | Buy on Amazon |
| 4 |
|
A Course in Game Theory (Mit Press) | $55.99 | Buy on Amazon |
| 5 |
|
Game Theory: 100+ Exercises in Strategy and Decision with Detailed Solutions to Master Nash... | $25.99 | Buy on Amazon |
As an Amazon Associate I earn from qualifying purchases.
The formal strategic-form test
Suppose Game A has strategy sets S1, S2, … and Game B has corresponding sets Ŝ1, Ŝ2, …. Under the definition used by Alvin E. Roth and J. Keith Murnighan, the games are strategically equivalent when, for every player i, there is a transformation Ti: Si → Ŝi that is both one-to-one and onto, and the outcome/payoff association agrees for every strategy profile after applying those transformations. See the formal definition in Roth and Murnighan’s paper.
“One-to-one” means that two different strategies in Game A cannot map to the same strategy in Game B. “Onto” means that every strategy in Game B is reached by the mapping. Together these properties make a bijection: a complete relabeling with no mergers or omissions. If the sets are finite, equal strategy counts are necessary for a bijection, but equal counts alone say nothing about whether payoffs line up.
#1 Best Overall
What must be preserved
For every joint profile s = (s1, …, sn) in Game A, construct the corresponding profile T(s) = (T1(s1), …, Tn(sn)) in Game B. The relevant outcome and each player’s payoff or preference relationship must match under the equivalence standard you selected. A single unmatched profile defeats that particular proposed mapping.
This criterion allows harmless changes in labels such as “left” becoming “right” or “cooperate” becoming “option 1,” provided the relabeling is bijective and the complete strategic structure follows it. It does not allow deleting a strategy, combining two strategies into one, or checking only the profile that happens to be an equilibrium.
Choose the preservation target before comparing
Write down what your conclusion is supposed to establish. A comparison that preserves payoff matrices may not preserve the game tree; one that preserves preference rankings may not preserve exact utility magnitudes.
Free tools Windows power users keep installed
One-click scans. No signup required.
| Axis | What to inspect | Why it matters |
|---|---|---|
| Players | Whether the same players are being compared, or an explicit player correspondence is supplied | Player identity determines whose strategies and preferences are evaluated. |
| Strategy sets | A one-to-one and onto map for each player | Bijections are part of the cited strategic-equivalence definition. |
| Profile mapping | Every joint strategy profile after the proposed maps | The criterion quantifies over all profiles, not a sample. |
| Outcomes and payoffs | The outcome function and each player’s payoff association | These are the strategic consequences attached to profiles. |
| Preference interpretation | Exact payoff equality or preservation of preference rankings | Utility functions represent preferences, and different equivalence notions permit different numerical changes. |
| Timing and information | Move order, information sets, observations, and transitions | Strategic form can omit sequential and informational properties. |
| Intended conclusion | The prediction or solution property you want to transfer | Only properties invariant under the chosen equivalence are licensed by the comparison. |
The Stanford Encyclopedia of Philosophy describes strategic games in terms of players, outcomes, strategy sets, an outcome function, and preferences; its discussion of utility explains how numerical functions can represent preference relations. Use that distinction when deciding whether “same payoffs” means identical numbers or merely the same ordering of outcomes. See the Stanford Encyclopedia’s account of normal-form games and utility.
Strategic form and extensive form are different comparison problems
What a normal form records
A strategic or normal form lists each player’s complete strategies and the outcome or payoff associated with every joint choice. A matrix is a convenient display for two-player finite games. If the question is only which complete strategies lead to which outcomes and preferences, a bijective profile-by-profile comparison is often the right test.
Rank #2
What a normal form can omit
An extensive-form game is represented as a tree. It records the sequence of moves, which player moves at each node, what a player knows when choosing, information sets, and observations or transitions. Two sequential games can produce the same strategic-form table while differing in when choices occur or what players know. Those differences can affect questions about credibility, information, or sequential behavior.
The Stanford Encyclopedia of Philosophy’s game-theory overview explains why strategic form is appropriate when order is irrelevant and extensive form is needed when order and information matter. Therefore, a normal-form match supports a claim about the represented strategic possibilities, not automatically a claim that the original game trees are interchangeable.
Structural notions beyond payoff-table matching
For richer comparisons, state a structural criterion such as isomorphism or bisimulation. The Stanford game-equivalence handout presents equivalence as a family of structural and semantic ideas and discusses transformations of extensive games, including cases involving contiguous moves. Treat those results as criteria for particular models and claims, not as a universal replacement for the strategic-form test.
A step-by-step comparison procedure
- Identify the representation. Label each object as a strategic/normal-form game, an extensive-form game, or another formal model. Record whether strategies are pure complete plans, contingent plans, or a different kind of object.
- Name the players and any allowed player correspondence. If player identities are fixed, compare player
iwith playeri. If the problem allows renaming players, write that permutation explicitly and apply it consistently to strategies, outcomes, and preferences. - State the equivalence target. Decide whether you need to preserve strategy availability, outcome labels, exact payoff numbers, preference rankings, timing, information, observations, or a solution property. Do not switch from one target to another midway through the argument.
- List every strategy set. For each player, write the complete set in both games. Include strategies that are never optimal or never used in an equilibrium; they still count for the bijection and for profile checking.
- Propose candidate bijections. Map each strategy in Game A to exactly one strategy in Game B, verify that no two map to the same target, and verify that every target strategy is reached. If finite sets have different cardinalities, stop: the cited bijective condition cannot hold.
- Map every profile. For each profile in Game A, apply all player-specific maps. In a two-player game this means checking every cell; in an
n-player game it means checking the Cartesian product of the strategy sets. - Compare outcomes and payoffs under the selected rule. Check the outcome identity or outcome correspondence, then each player’s payoff or preference relation. Record the first mismatch rather than averaging discrepancies or ignoring an inconvenient cell.
- Try alternative relabelings when labels are arbitrary. A failed mapping does not prove none exists if several bijections are possible. Systematically test the remaining permutations or use an isomorphism search for larger finite games.
- Check sequential structure separately when relevant. Compare the game trees, move order, information sets, observations, and transitions. A successful table comparison does not settle this step.
- State exactly what the result licenses. Say “equivalent under the specified strategic-form criterion” or identify the structural criterion used. Then name which predictions or analyses can be transferred and which features were not compared.
Worked examples
Example 1: a pure relabeling that passes
Consider two two-player games. In Game A, each player has strategies U and D; in Game B, the corresponding strategies are Left and Right. Let the mapping be U → Left and D → Right for both players. The payoff tables are:
| Game A | U | D |
|---|---|---|
| U | (3, 2) | (0, 0) |
| D | (0, 0) | (2, 3) |
| Game B | Left | Right |
|---|---|---|
| Left | (3, 2) | (0, 0) |
| Right | (0, 0) | (2, 3) |
There are bijections for both players, and all four mapped cells have the same payoff vectors. Under the cited expanded strategic-form definition, this is strategic equivalence. The labels changed; the profile-to-payoff association did not.
Rank #3
Example 2: equal strategy counts but no equivalence under the proposed map
Now keep the same two strategies per player but change one cell in Game B:
The Tool Desk
Outbyte Driver Updater FREEFix the driver behind crashes, sound loss and screen glitchesFind Drivers →Outbyte PC Repair FREEClear out junk files and repair common Windows errorsFree Scan →| Game A | U | D |
|---|---|---|
| U | (3, 2) | (0, 0) |
| D | (0, 0) | (2, 3) |
| Game B | Left | Right |
|---|---|---|
| Left | (3, 2) | (0, 0) |
| Right | (0, 0) | (3, 2) |
The strategy counts match, but under U → Left and D → Right, the lower-right profile has different payoffs. That mapping fails. You must test the other possible relabelings before concluding that no bijection works; if every bijection leaves a mismatch, the games are not equivalent under the exact-payoff criterion.
Example 3: preference preservation versus exact utility preservation
Suppose one game assigns a player utilities 1, 2, and 4 to three outcomes, while another assigns 10, 20, and 40 to the corresponding outcomes. The order is the same, so the numbers can represent the same preference ranking if the chosen model treats utilities ordinally. However, the values are not equal. A definition that requires the payoff association itself to agree will reject the pair; a preference-based criterion may accept it. State the criterion instead of assuming that every positive rescaling is harmless.
How to check a comparison systematically
For a finite game, make a worksheet with one row per profile. Include the original profile, its mapped profile, Game A’s outcome and payoff vector, Game B’s outcome and payoff vector, and a pass/fail column for each required property.
| Profile in A | Mapped profile in B | Outcome match | Player 1 match | Player 2 match |
|---|---|---|---|---|
(s1, s2) |
(T1(s1), T2(s2)) |
Yes/no | Yes/no | Yes/no |
For more than two players, generate the Cartesian product of all strategy sets. A simple algorithm is:
Rank #4
- Enumerate every profile in Game A.
- Apply the candidate strategy map component by component.
- Look up the outcome and payoff or preference data in both games.
- Reject the candidate map on the first required mismatch.
- If it passes all profiles, report equivalence only for the stated criterion.
When labels have no meaning, enumerate possible strategy permutations for each player. The search grows quickly as the number of strategies increases, so use canonical ordering or an isomorphism algorithm for large finite tables. The logical requirement does not change: every player’s map must be bijective, and every profile must satisfy the preservation rule.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Common mistakes and how to correct them
Checking only the equilibrium
An equilibrium is one profile or a small set of profiles. Strategic equivalence requires the full profile-to-outcome/payoff relation. Two games can share an equilibrium while differing at other profiles and therefore fail the equivalence test.
Treating equal matrix dimensions as proof
Two-by-two tables have equal cardinalities, but their entries may encode different incentives. Use cardinality only as an initial necessary check, then compare all cells under candidate bijections.
Allowing a many-to-one strategy map
If two strategies in Game A are merged into one in Game B, the map is not one-to-one. If a strategy in Game B is never reached, it is not onto. Either defect violates the cited strategic-equivalence condition.
Quick wins for a faster PC:
Scan for outdated or missing drivers - takes under a minuteDriver Scan →Repair Windows errors before they cause bigger problemsFix Now →Assuming utility transformations are always allowed
Whether a positive affine or monotone transformation is acceptable depends on the model and purpose. If the definition preserves exact payoff associations, changed numbers fail that test. If the purpose is only preference ordering, compare rankings and state that weaker criterion.
Ignoring outcomes because payoffs look similar
Some definitions distinguish outcomes from the utilities assigned to them. If outcome identity matters, preserve the outcome correspondence as well as the payoff vector. Do not infer that equal utilities imply identical outcomes.
Calling sequential games equivalent from matrices alone
A normal-form table may hide who moved first, what a player observed, or which choices were available at each node. When those facts matter, compare the extensive forms and identify the structural notion being used.
Transferring a conclusion too broadly
A strategic-form bijection can justify transferring properties that depend only on the preserved strategic data. It does not automatically preserve claims about information sets, timing, or behavior at unreached nodes unless those features are part of the chosen equivalence.
How to report the result
A precise write-up should contain five elements:
- Representation: identify whether the objects are strategic form, extensive form, or another model.
- Player correspondence: state whether players are fixed or relabeled.
- Strategy maps: list the bijection for every player and verify one-to-one and onto properties.
- Preservation check: explain that every profile was tested and specify whether outcomes, exact payoffs, or preference rankings were compared.
- Scope: state whether timing, information, observations, and transitions were also preserved.
For example: “Under the exact-payoff strategic-form criterion, the games are equivalent via the listed bijections; all 16 profiles have matching outcome and payoff associations. The comparison does not establish equivalence of their extensive forms because move order and information sets were not matched.” That wording separates a supported conclusion from a broader claim the test did not address.
A reusable checklist
- Have you named the game representation?
- Have you identified the players or supplied an explicit player permutation?
- Does each player have a one-to-one, onto strategy map?
- Are all strategy profiles included, including profiles that are not equilibria?
- Are outcomes compared separately from payoff numbers?
- Have you stated whether exact utilities or preference rankings matter?
- Have you tested alternative relabelings where labels are arbitrary?
- If the games are sequential, have you compared move order and information sets?
- Does your final claim name the equivalence criterion and its limits?
Further reading
For the strategic-versus-extensive distinction and the role of sequence and information, consult the Stanford Encyclopedia of Philosophy’s Game Theory entry. For the expanded strategic-form definition with bijective strategy transformations, read Roth and Murnighan’s paper. For structural ideas such as isomorphism and bisimulation in game descriptions, see the Stanford game-equivalence handout. For the distinction between strategic-game forms, outcomes, preferences, and utility representations, see the Stanford Encyclopedia discussion of logic and power in normal-form games.
Quick Recap
Product prices and availability are accurate as of the date/time indicated and are subject to change. Any price and availability information displayed on Amazon at the time of purchase will apply.
